Effects of Cumulative Practice on Mathematics Problem-Solving Behavior
Kristin H. Hazlett
Mathematics education has long been in need of improved methods of instruction, particularly in
the area of problem-solving skills. This study compared three methods of training rules about
laws of exponents and order of operations. All three training methods used the same mastery
criterion for training each rule and included the same number of practice trials during review
sessions that preceded each test. The difference between conditions involved what types of
problems were presented during the reviews. For each review session, the cumulative group (n =
11) practiced 50 problems covering all rules learned up to that review. The simple review group
(n = 11) practiced 50 problems on one previous rule, and the extra practice group (n = 11)
practiced 50 more problems of the same rule they had just mastered. Tests were administered
after each review.
Though no initial differences existed between groups on any measure, the last test revealed that
the cumulative group scored significantly higher than the other groups on items that involved
novel applications of the individual rules. Moreover, the cumulative group outperformed the
other two groups on untrained, complex problem-solving tasks that required novel combinations
of the individual rules. In addition, the cumulative group performed the problem-solving tasks at
a significantly faster rate than the other groups. There were no statistical differences among
groups on a retention test, however, which was partially due to a reduction in sample size, as
well as increases in variability of performance within groups.
Overall, the findings support the viewpoints of behavioral educators that mastery of component
skills facilitates performance on higher-level skills and that novel behavior is fundamentally
related to its component parts. The results also extend the research of behavioral educators by
removing the confounded variables of simple review and extra practice found in previous studies
and by showing the effects of cumulative practice on problem-solving behavior. Finally, the
results suggest that an approach to training problem solving similar to the one presented in this
study may yield higher levels of success than methods used by traditional mathematics
educators.
This research was partially funded by dissertation grants from the Office of Academic Affairs
and the Psychology Alumni Fund at West Virginia University.